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Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

Numerical Algorithms volumeΒ 69,Β pages 253–270 (2015)Cite this article

Abstract

We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ΞΆ(s, a) for \(s, a \in \mathbb {C}\), along with an arbitrary number of derivatives with respect to s, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.

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Authors and Affiliations

  1. RISC, Johannes Kepler University, 4040, Linz, Austria

    Fredrik Johansson

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  1. Fredrik Johansson
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Correspondence to Fredrik Johansson.

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Johansson, F. Rigorous high-precision computation of the Hurwitz zeta function and its derivatives. Numer Algor 69, 253–270 (2015). https://doi.org/10.1007/s11075-014-9893-1

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  • DOI: https://doi.org/10.1007/s11075-014-9893-1

Keywords

  • Hurwitz zeta function
  • Riemann zeta function
  • Arbitrary-precision arithmetic
  • Rigorous numerical evaluation
  • Fast polynomial arithmetic
  • Power series

PAC Codes

  • 65D20
  • 68W30
  • 33F05
  • 11-04
  • 11M06
  • 11M35

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